Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE$_2$

Abstract

This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected grid domain and conditioned to exit it through another boundary vertex specified in advance. The primary contribution of the present paper is an estimate, which states that the excursion of the conditioned walk leaves an intrinsic neighborhood of its initial point not 'along' the boundary but through an intrinsic interior of the domain with high probability. Based on this result we give a proof for the convergence to the chordal SLE, a result that has recently been proved by Suzuki [12] under an analyticity assumption on the boundary of the domain arising in the limit.